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In-Depth Information
Hidden MRF
y
1
=1
y
2
=1
Must-link (
c
=1)
12
Cannot−link
(
c
=
−
1)
23
Must-link (
c
14
=1)
y
5
=3
y
4
=1
y
3
=2
x
1
x
2
x
4
x
5
x
3
Observed data
FIGURE 7.8
: A hidden Markov random field.
C
Θ
Y
X
FIGURE 7.9
: Graphical plate model of variable dependence.
model can be factorized as follows:
P(
X, Y,
Θ
|
C
)=P(Θ
|
C
)P(
Y
|
Θ
,C
)P(
X
|
Y,
Θ
,C
)
(7.15)
The graphical plate model (10) of the dependence between the random
variables in the HMRF is shown in Figure 7.9. The prior probability of Θ is
assumed to be independent of
C
,sothatP(Θ
C
) = P(Θ). The probability of
observing the label configuration
Y
depends on the constraints
C
and current
generative model parameters Θ. Observed data instances corresponding to
variables
X
are generated using the model parameters Θ based on cluster
labels
Y
, independent of the constraints
C
.Thevariables
X
are assumed to
be mutually independent: each
x
i
is generated individually from a conditional
probability distribution P(
x
i
|
|
y
i
,
Θ).
Basu et al. (6) show that the joint probability on the HMRF is equivalent
to maximizing:
C
)=P(Θ)
1
v
(
i, j
)
n
y
i
,
Θ)
Z
exp
−
P(
X, Y,
Θ
|
p
(
x
i
|
(7.16)
c
ij
∈C
i
=1
They chose the following Gibbs potential for P(
Y
|
Θ
,C
):
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